A raidator in need of a flush will not cause a battery drain.
A short in the wiring.
Shorted diodes in the alternator.
dead cell in the battery or something is on pulling power from the battery.
Your alternator could be bad.
Any relay could be the cause.
If no light is on and no relay is stuck then the battery has a dead cell and needs replacing.
Could be a shorted wire, or bad battery.
I'd recommend you do a battery test and charging system test. The battery would be my first suspect.
Good possibility
Yes possible. If switch does not open on engine shut down battery will drain If switch does not close on start up battery will not charge
Yes, if the relay was stuck it would drain the battery.